\begin{itemize}

\item There are node-edge incidence matrices of undirected graphs that are totally unimodular, for example the $K_2$: $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$, where the determinants of both square submatrices equal 1. It can easily be shown that the node-edge incidence matrix of every graph containing circuits of odd length is not totally unimodular, whereas the node-edge incidence matrix of every bipartite graph is totally unimodular.

\item Node-arc incidence matrices of directed graphs are totally unimodular. We prove this by verifying the sufficient condition from Proposition 47:

\begin{enumerate}[(i)]
\item A node-arc incidence matrix contains only the values \{0, 1, -1\} by definition.
\item Each column corresponds to one arc and thus contains exactly one entry of value 1 and one entry of value -1.
\item Since every column sums to zero, partitioning the set of rows into the set of all rows and the empty set fulfills the property.
\end{enumerate}

\end{itemize}